There are 210 households in a certain residential complex. All househo

Question

There are 210 households in a certain residential complex. All households in the complex that have more than one dog also have at least one cat. All households that have at least one cat have a pet rodent. How many households in the complex have a pet rodent?

(1) 18 households in the complex have two or more dogs.

(2) 90% of households in the complex that have a pet rodent have at least one cat.

(1) 18 households in the complex have two or more dogs.

(2) 90% of households in the complex that have a pet rodent have at least one cat.

gmatophobia · Accepted Answer

A good CR-type question!

Statement 1

(1) 18 households in the complex have two or more dogs.

Based on the given information, we can deduce that a minimum of 18 households within the complex possess rodents as pets. This conclusion is drawn from Statement 1, which specifies that 18 households in the complex own two or more dogs. Additionally, considering the initial question premise, it is established that every household in the complex with multiple dogs also harbors at least one cat, and each household with a cat possesses a pet rodent. Consequently, we can affirm that the complex comprises a minimum of 18 households that own pet rodents.

However, we cannot conclude that ONLY 18 households have a pet rodent. The complex may include households with solely pet rodents or those with a combination of a pet rodent and one dog. Therefore, the statement alone is insufficient to definitively answer the question, leading us to eliminate options A and D.

Statement 2

(2) 90% of households in the complex that have a pet rodent have at least one cat

This statement furnishes details regarding households that possess a pet rodent, explicitly stating that all such households also possess a cat. Nevertheless, the initial information about the total number of households with a pet rodent is absent. Without this essential data, we lack the necessary foundation to find answers to the question asked. Therefore, this statement is insufficient.

Combined

The combined statements fail to provide a conclusive answer as they do not eliminate the possibility that a household might exclusively have a rodent as a pet or could have another animal in addition to a rodent. The ambiguity persists, preventing us from definitively determining the composition of households with pet rodents.

Hence,  Option E

oiploiloi · Answer

I think the answer is E, both options are insufficient.

From the argument, we know that:
- (household has > 1 dog) = (household has >= 1 cat)
- (household has >= 1 cat) = (household has >= 1 rodent)

Question: how many households have a rodent

Option 1 
18 households have 2 or more dogs i.e 18 households have > 1 dog

18 households with > 1 dog = 18 households with >= 1 cat
But what about all the households with cats but no dog or just one dog? They are not reflected in the 18 households mentioned above.
Therefore it is more accurate to say that:
(18 households with > 1 dog) = (18 or more households with >= 1 cat)
(18 or more households with >= 1 cat) = (18 or more households with rodents)

18 or more can be 18, 19, 29, 49....
Because we don't have one distinct possible answer, this option is insufficient

Option 2 
90% of households in the complex that have a pet rodent have at least one cat

You can't calculate anything from this. Clearly insufficient.

Together 
Combining doesn't really add any new information to make this question solvable

KarishmaB · Answer

Make a Venn diagram Screenshot 2025-01-30 at 10.21.43 AM.png (1) 18 households in the complex have two or more dogs. The 2 dogs circle has 18 households. No info on rodent circle. Not sufficient (2) 90% of households in the complex that have a pet rodent have at least one cat. Cat circle has 90% of recent circle households. But how many households have rodents, no info. Not sufficient Using both together, we do not know the number of households in rodent circle still. The relation between 2 dog circle and cat circle is not available. Screenshot 2025-01-30 at 10.25.55 AM.png Answer (E)

Samé · Answer

KarishmaB

Please, why is it not possible to establish from the 2 statements that 90% of rodent households = 18 ?

KarishmaB · Answer

90% of the rodents household are cats households. 
18 is the number of 2 or more dogs households. There could be some cats households that do not have a dog or have only 1 dog. 
Hence the number of cats households are at least 18 is all I know. They could easily be more than 18.

ManifestDreamMBA · Answer

Solved it like a CR question:

Some dog = Some cat
Some cat = One rodent
This means Some dog = One rodent

Ways to own rodent
1. have some dog
2. have a cat
3. have a rodent

Statment 1:
This just tells me 18 houses have #1. We don't know about 2 and 3. Insufficient

Statement 2:
This just tells me of the houses which have rodents, 90% have at least one cat. This doesn't tell me the total number of cats just the distribution of rodent owners who also have a cat. Insufficient

Combined:
I still just have a number for rodents who are also a dog owner (i.e. #1 above). Insufficient

Answer is E

egmat · Answer

Understanding What We're Given: 
You have 210 households total, and there's an important chain of logic here:
- If a household has >1 dog → it MUST have at least one cat
- If a household has at least one cat → it MUST have a rodent

So we have this subset relationship:  	ext{Multi-dog households} \subseteq 	ext{Cat households} \subseteq 	ext{Rodent households}

The question asks for the  exact  number of households with rodents - remember, in DS, we need a unique value!

Analyzing Statement (1):  18 households have two or more dogs

Let's think about what this tells us. Since these 18 households must have cats (from our given info), and households with cats must have rodents, we know  at least  18 households have rodents.

But here's what you need to see - this doesn't tell us:
- How many households might have cats but only one dog (or no dogs)
- How many households might have rodents but no cats

We could have exactly 18 households with rodents, or 50, or 100... Statement 1 is  NOT sufficient .

Analyzing Statement (2):  90% of rodent households have at least one cat

This gives us a proportional relationship. If we call the number of rodent households R and cat households C, then  C = 0.9R .

But without knowing either C or R, we can't determine the exact value. If R = 100, then C = 90. If R = 200, then C = 180. Both work! Statement 2 is  NOT sufficient .

Combining Both Statements: 
Now here's where it gets interesting! We know:
- At least 18 households have cats (from Statement 1)
- Cat households = 90% of rodent households (from Statement 2)

This gives us  18 \leq 0.9R , so  R \geq 20 .

But notice - we still don't have an exact value! Let me show you two valid scenarios:
- If exactly 18 households have cats:  18 = 0.9R  →  R = 20  ✓
- If 27 households have cats:  27 = 0.9R  →  R = 30  ✓

Since we can have different valid answers (20, 30, or other values), even combining both statements is  NOT sufficient .

Answer: E

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You can check out the  step-by-step solution on Neuron by e-GMAT  to master the systematic framework for handling complex set relationships in Data Sufficiency questions. You can also explore other GMAT official questions with detailed solutions on Neuron for structured practice  here . The full solution reveals a powerful technique for visualizing these subset relationships that applies to many similar DS questions you'll encounter.

There are 210 households in a certain residential complex. All househo

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